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Properties

Label 2-490-245.103-c1-0-11
Degree $2$
Conductor $490$
Sign $0.701 - 0.712i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.593 + 0.804i)2-s + (0.858 − 0.162i)3-s + (−0.294 − 0.955i)4-s + (2.23 + 0.0576i)5-s + (−0.379 + 0.787i)6-s + (2.57 − 0.601i)7-s + (0.943 + 0.330i)8-s + (−2.08 + 0.816i)9-s + (−1.37 + 1.76i)10-s + (−2.11 + 5.38i)11-s + (−0.408 − 0.772i)12-s + (0.808 + 0.0911i)13-s + (−1.04 + 2.43i)14-s + (1.92 − 0.313i)15-s + (−0.826 + 0.563i)16-s + (3.43 + 0.128i)17-s + ⋯
L(s)  = 1  + (−0.419 + 0.568i)2-s + (0.495 − 0.0938i)3-s + (−0.147 − 0.477i)4-s + (0.999 + 0.0258i)5-s + (−0.154 + 0.321i)6-s + (0.973 − 0.227i)7-s + (0.333 + 0.116i)8-s + (−0.693 + 0.272i)9-s + (−0.434 + 0.557i)10-s + (−0.637 + 1.62i)11-s + (−0.117 − 0.223i)12-s + (0.224 + 0.0252i)13-s + (−0.279 + 0.649i)14-s + (0.498 − 0.0809i)15-s + (−0.206 + 0.140i)16-s + (0.831 + 0.0311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51123 + 0.632949i\)
\(L(\frac12)\) \(\approx\) \(1.51123 + 0.632949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.593 - 0.804i)T \)
5 \( 1 + (-2.23 - 0.0576i)T \)
7 \( 1 + (-2.57 + 0.601i)T \)
good3 \( 1 + (-0.858 + 0.162i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (2.11 - 5.38i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-0.808 - 0.0911i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (-3.43 - 0.128i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-3.76 + 6.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0216 + 0.579i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (0.638 - 0.145i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-4.49 + 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.82 - 4.66i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (-1.30 - 2.71i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-3.04 - 8.71i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (7.03 + 5.19i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (-0.513 - 0.271i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.490 + 6.54i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-3.33 + 10.8i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (8.69 + 2.33i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.12 - 4.91i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.08 + 0.801i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (7.93 + 4.58i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.01 + 8.96i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-5.80 - 14.7i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (2.71 + 2.71i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.84043291706953893011868703044, −9.948286623927957005830541663774, −9.299437458862110016458435213798, −8.266221639500842667128091860235, −7.58256006552622139972826853625, −6.65786719787560946733723839319, −5.30139791419065195980701176523, −4.82074646660582928131934958096, −2.75094325937812659548909816938, −1.62519323253577074974915699505, 1.32344183931414646332502858858, 2.69269828809041374252038544479, 3.58553885258508585528969867060, 5.42379760393344915421948609283, 5.84462290515371544118470621139, 7.61528901525165711753307458652, 8.484900822879520286317495093751, 8.897265351399655597803189502828, 10.05866537045448691312840843490, 10.70192493290821934374264455022

Graph of the $Z$-function along the critical line